Central Limit Theory for Combined Cross-Section and Time Series with an Application to Aggregate Productivity Shocks

TL;DR

This paper develops a central limit theorem for combined cross-section and time series data that accounts for dependence via common factors, enabling more accurate inference in empirical economics involving aggregate shocks.

Contribution

It introduces a novel CLT explicitly modeling dependence between cross-sectional and time series data through common factors, improving asymptotic analysis of parameter estimates.

Findings

Established asymptotic properties of parameter estimates with combined data.

Provided a practical implementation similar to standard formulas.

Enhanced understanding of dependence effects in empirical models.

Abstract

Combining cross-section and time series data is a long and well established practice in empirical economics. We develop a central limit theory that explicitly accounts for possible dependence between the two data sets. We focus on common factors as the mechanism behind this dependence. Using our central limit theorem (CLT) we establish the asymptotic properties of parameter estimates of a general class of models based on a combination of cross-sectional and time series data, recognizing the interdependence between the two data sources in the presence of aggregate shocks. Despite the complicated nature of the analysis required to formulate the joint CLT, it is straightforward to implement the resulting parameter limiting distributions due to a formal similarity of our approximations with the standard Murphy and Topel's (1985) formula.